Optimal. Leaf size=187 \[ -\frac {a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{\sqrt {c}}+\frac {\sqrt {b} \left (15 a^2 d^2-10 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{8 d^{5/2}}-\frac {b \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-7 a d)}{8 d^2}+\frac {b \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{4 d} \]
________________________________________________________________________________________
Rubi [A] time = 0.23, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {446, 102, 154, 157, 63, 217, 206, 93, 208} \begin {gather*} \frac {\sqrt {b} \left (15 a^2 d^2-10 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{8 d^{5/2}}-\frac {a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{\sqrt {c}}-\frac {b \sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-7 a d)}{8 d^2}+\frac {b \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{4 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 93
Rule 102
Rule 154
Rule 157
Rule 206
Rule 208
Rule 217
Rule 446
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^{5/2}}{x \sqrt {c+d x^2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^{5/2}}{x \sqrt {c+d x}} \, dx,x,x^2\right )\\ &=\frac {b \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{4 d}+\frac {\operatorname {Subst}\left (\int \frac {\sqrt {a+b x} \left (2 a^2 d-\frac {1}{2} b (3 b c-7 a d) x\right )}{x \sqrt {c+d x}} \, dx,x,x^2\right )}{4 d}\\ &=-\frac {b (3 b c-7 a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{8 d^2}+\frac {b \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{4 d}+\frac {\operatorname {Subst}\left (\int \frac {2 a^3 d^2+\frac {1}{4} b \left (3 b^2 c^2-10 a b c d+15 a^2 d^2\right ) x}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx,x,x^2\right )}{4 d^2}\\ &=-\frac {b (3 b c-7 a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{8 d^2}+\frac {b \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{4 d}+\frac {1}{2} a^3 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx,x,x^2\right )+\frac {\left (b \left (3 b^2 c^2-10 a b c d+15 a^2 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx,x,x^2\right )}{16 d^2}\\ &=-\frac {b (3 b c-7 a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{8 d^2}+\frac {b \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{4 d}+a^3 \operatorname {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x^2}}{\sqrt {c+d x^2}}\right )+\frac {\left (3 b^2 c^2-10 a b c d+15 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x^2}\right )}{8 d^2}\\ &=-\frac {b (3 b c-7 a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{8 d^2}+\frac {b \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{4 d}-\frac {a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{\sqrt {c}}+\frac {\left (3 b^2 c^2-10 a b c d+15 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x^2}}{\sqrt {c+d x^2}}\right )}{8 d^2}\\ &=-\frac {b (3 b c-7 a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{8 d^2}+\frac {b \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{4 d}-\frac {a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{\sqrt {c}}+\frac {\sqrt {b} \left (3 b^2 c^2-10 a b c d+15 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{8 d^{5/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.67, size = 213, normalized size = 1.14 \begin {gather*} \frac {1}{8} \left (-\frac {8 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{\sqrt {c}}+\frac {\left (-15 a^3 d^3+25 a^2 b c d^2-13 a b^2 c^2 d+3 b^3 c^3\right ) \sqrt {\frac {b \left (c+d x^2\right )}{b c-a d}} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b c-a d}}\right )}{d^{5/2} \sqrt {c+d x^2} \sqrt {b c-a d}}+\frac {b \sqrt {a+b x^2} \sqrt {c+d x^2} \left (9 a d-3 b c+2 b d x^2\right )}{d^2}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 2.27, size = 276, normalized size = 1.48 \begin {gather*} -\frac {a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c+d x^2}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c}}+\frac {\left (15 a^2 \sqrt {b} d^2-10 a b^{3/2} c d+3 b^{5/2} c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {d} \sqrt {a+b x^2}}\right )}{8 d^{5/2}}-\frac {b \sqrt {c+d x^2} \left (\frac {7 a^2 b d^2 \left (c+d x^2\right )}{a+b x^2}-9 a^2 d^3+\frac {3 b^3 c^2 \left (c+d x^2\right )}{a+b x^2}-\frac {10 a b^2 c d \left (c+d x^2\right )}{a+b x^2}+14 a b c d^2-5 b^2 c^2 d\right )}{8 d^2 \sqrt {a+b x^2} \left (d-\frac {b \left (c+d x^2\right )}{a+b x^2}\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 10.42, size = 1075, normalized size = 5.75 \begin {gather*} \left [\frac {8 \, a^{2} d^{2} \sqrt {\frac {a}{c}} \log \left (\frac {{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{4} + 8 \, a^{2} c^{2} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x^{2} - 4 \, {\left (2 \, a c^{2} + {\left (b c^{2} + a c d\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {\frac {a}{c}}}{x^{4}}\right ) + {\left (3 \, b^{2} c^{2} - 10 \, a b c d + 15 \, a^{2} d^{2}\right )} \sqrt {\frac {b}{d}} \log \left (8 \, b^{2} d^{2} x^{4} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x^{2} + 4 \, {\left (2 \, b d^{2} x^{2} + b c d + a d^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {\frac {b}{d}}\right ) + 4 \, {\left (2 \, b^{2} d x^{2} - 3 \, b^{2} c + 9 \, a b d\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{32 \, d^{2}}, \frac {4 \, a^{2} d^{2} \sqrt {\frac {a}{c}} \log \left (\frac {{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{4} + 8 \, a^{2} c^{2} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x^{2} - 4 \, {\left (2 \, a c^{2} + {\left (b c^{2} + a c d\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {\frac {a}{c}}}{x^{4}}\right ) - {\left (3 \, b^{2} c^{2} - 10 \, a b c d + 15 \, a^{2} d^{2}\right )} \sqrt {-\frac {b}{d}} \arctan \left (\frac {{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {-\frac {b}{d}}}{2 \, {\left (b^{2} d x^{4} + a b c + {\left (b^{2} c + a b d\right )} x^{2}\right )}}\right ) + 2 \, {\left (2 \, b^{2} d x^{2} - 3 \, b^{2} c + 9 \, a b d\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{16 \, d^{2}}, \frac {16 \, a^{2} d^{2} \sqrt {-\frac {a}{c}} \arctan \left (\frac {{\left ({\left (b c + a d\right )} x^{2} + 2 \, a c\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {-\frac {a}{c}}}{2 \, {\left (a b d x^{4} + a^{2} c + {\left (a b c + a^{2} d\right )} x^{2}\right )}}\right ) + {\left (3 \, b^{2} c^{2} - 10 \, a b c d + 15 \, a^{2} d^{2}\right )} \sqrt {\frac {b}{d}} \log \left (8 \, b^{2} d^{2} x^{4} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x^{2} + 4 \, {\left (2 \, b d^{2} x^{2} + b c d + a d^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {\frac {b}{d}}\right ) + 4 \, {\left (2 \, b^{2} d x^{2} - 3 \, b^{2} c + 9 \, a b d\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{32 \, d^{2}}, \frac {8 \, a^{2} d^{2} \sqrt {-\frac {a}{c}} \arctan \left (\frac {{\left ({\left (b c + a d\right )} x^{2} + 2 \, a c\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {-\frac {a}{c}}}{2 \, {\left (a b d x^{4} + a^{2} c + {\left (a b c + a^{2} d\right )} x^{2}\right )}}\right ) - {\left (3 \, b^{2} c^{2} - 10 \, a b c d + 15 \, a^{2} d^{2}\right )} \sqrt {-\frac {b}{d}} \arctan \left (\frac {{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {-\frac {b}{d}}}{2 \, {\left (b^{2} d x^{4} + a b c + {\left (b^{2} c + a b d\right )} x^{2}\right )}}\right ) + 2 \, {\left (2 \, b^{2} d x^{2} - 3 \, b^{2} c + 9 \, a b d\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{16 \, d^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.02, size = 446, normalized size = 2.39 \begin {gather*} \frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \left (-8 \sqrt {b d}\, a^{3} d^{2} \ln \left (\frac {a d \,x^{2}+b c \,x^{2}+2 a c +2 \sqrt {a c}\, \sqrt {x^{4} b d +a d \,x^{2}+b c \,x^{2}+a c}}{x^{2}}\right )+15 \sqrt {a c}\, a^{2} b \,d^{2} \ln \left (\frac {2 b d \,x^{2}+a d +b c +2 \sqrt {x^{4} b d +a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )-10 \sqrt {a c}\, a \,b^{2} c d \ln \left (\frac {2 b d \,x^{2}+a d +b c +2 \sqrt {x^{4} b d +a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+3 \sqrt {a c}\, b^{3} c^{2} \ln \left (\frac {2 b d \,x^{2}+a d +b c +2 \sqrt {x^{4} b d +a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )+4 \sqrt {x^{4} b d +a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}\, \sqrt {a c}\, b^{2} d \,x^{2}+18 \sqrt {x^{4} b d +a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}\, \sqrt {a c}\, a b d -6 \sqrt {x^{4} b d +a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}\, \sqrt {a c}\, b^{2} c \right )}{16 \sqrt {x^{4} b d +a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}\, \sqrt {a c}\, d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (b\,x^2+a\right )}^{5/2}}{x\,\sqrt {d\,x^2+c}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b x^{2}\right )^{\frac {5}{2}}}{x \sqrt {c + d x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________